Can a Hunter Resolve Two Squirrels 10 cm Apart at 1.6 km Without a Scope?

[DMOC]

Homework Statement

A hunter who is a bit of a braggart claims that, from a distance of 1.6 km, he can selectively shoot either of two squirrels whoare sitting ten centimeters apart on the same branch of a tree. What's more, he claims that he can do this without the aid of a telescopic sight on his rifle. (a) Determine the diameter of the pupils of his eyes that would be required for him to be able to resolve the squrrels a separate objects. In this calculation, use a wavelength of 498 nm (in vacuum) for the light. (b) State whether his claim is reasonable and provide a reason for your answer. In evaluating his claim, consider that the human eye automatically adjusts the diameter of its pupil over a typical range of 2 to 8 mm, the larger values coming into play as the lighting becomes darker. note also that under dark conditions, the eye is most sensitive to a wavelength of 498 nm.

Homework Equations

minimum angle = 1.22(wavelength/diameter)

For the above equation, the minimum angle is in radians.

The Attempt at a Solution

I first tried to solve this by using the equation above. However, I ended up getting 2 unknowns, as demonstrated here:

angle=1.22(498 nm/diameter)

I need to find the angle and the diameter. I tried to use a right triangle diagram with 160000 cm as a leg and 10 cm as the other leg of the triange and used the tangent ratio to get the angle. However, this resulted in a very large diameter, so there's something I'm dong wrong here.

(This problem has to do with diffraction and Young's Double-Slit experiment.)

Any help would be appreciated at where I went wrong. This forum has been very helpful to me so far.

 

[lanedance]

Hi DMOC

I think you need to consider what the angle & diameter actually are...

the angle is the angle bewteen the objects you're trying to resolve, at the obsever... think how you might be able to find this from the info about the squirrels

the diameter, is the diameter of the observation device in this case the hunters eye

hope this helps

 

[DMOC]

So...I try to find the angle?

I attempted to do so and made a right triangle. The bottom leg is 160,000 cm and the squirrels are 10 cm apart, so that's the other leg. If I take the inverse tangent of those two, I get... 0.0035... degrees. Is that the minimum angle?

 

[lanedance]

sounding good - know where to go next?

note if you use radians for your angle
l = r.theta (arclength)
so you can just consider the circluar arc rather than a triangle, might simplify thiungs

when x<<1 they give the same answer as
tan(x) ~ x